A proof of the union-close set conjecture

Abstract

In this paper, we introduce the notion of the universe, induced communities, and cells with their corresponding spots. Using this language, we formulate and prove the union close set conjecture by showing that for any finite universe $\mathbb{U}$ and any induced community $\mathcal{M}_{\mathbb{U}}$ there exist some spot $a\in \mathbb{U}$ such that the density \begin{align} \mathcal{D}_{\mathcal{M}_{\mathbb{U}}}(a)\geq \frac{1}{2}.\nonumber \end{align}